Hybrid Discontinuous Galerkin Methods for Fluid Dynamics ...

Hybrid Discontinuous Galerkin Methods forFluid Dynamics and Solid Mechanics

Joachim Schoberl

Computational Mathematics in EngineeringInstitute for Analysis and Scientific Computing

Vienna University of Technology

Christoph Lehrenfeld

Center for Computational Engineering ScienceRWTH Aachen University

based on contributions bySabine Zaglmayr, Astrid Pechstein (born Sinwel)

Start project “hp-FEM”

Oberwolfach, Feb 2012

Joachim Schoberl

Incompressible flows

Stokes Equation:

Ω ⊂ Rd. Find velocity u ∈ [H1]d such that u = uD on ΓD, and pressure p ∈ Q := L2 such that∫Ω

∇u · ∇v +

∫Ω

div v p =

∫Ω

fv ∀ v ∈ V0

and incompressibility constraint ∫div u q = 0 ∀ q ∈ Q

with Dirichlet b.c. (no slip and inflow), point-wise mixed b.c. (slip) and Neumann (outflow).

Difficulty: Incompressibility constraint

Mixed finite elements: continuous pressure ? discontinuous pressure ? stabilized methods ?

Joachim Schoberl

Linear Elasticity

Ω ⊂ Rd. Find displacement u ∈ [H1]d such that u = uD on ΓD and∫Ω

Dε(u) : ε(v) =

∫Ω

fv ∀ v ∈ V0

with the linear strain operator ε(·) : [H1]d → [L2]d×d,sym

ε(u) =1

2

(∇u+ (∇u)T

)=(∂ui∂xj

+∂uj∂xi

)i,j=1,..d

and the isotropic material operator D : [L2]d×d → [L2]d×d

Dε = 2µε+ λ tr(ε)I

The stress tensor isσ = Dε(u)

Continuous and elliptic in [H1]d

BUT: Constants depend on λ/µ, and on the domain (Korn’s inequality) LOCKING !!

Joachim Schoberl Introduction

Von-Mises Stresses in a Machine Frame (linear elasticity)

Simulation with Netgen/NGSolve

45553 tets, p = 5, 3× 1074201 unknowns, 5 min on 8 core 2.4 GHz 64-bit PC 6 GB RAM


Toy Example: Sailplane

Incomp. N.-St., 2nd-order HDG elements, 59E3 elements, 1.65E6 dofs, 2GB RAM, 5 min (2-core 1.8GHz)


Function spaces H(curl) and H(div)

H(curl) = u ∈ [L2]d : curlu ∈ Ld×d,skew2 H(div) = u ∈ [L2]d : div u ∈ L2

Piece-wise smooth functions in

• H(curl) have continuous tangential components,

• H(div) have continuous normal components.

Important for constructing conforming finite elements such as Raviart Thomas, Brezzi-Douglas-Marini,and Nedelec elements.

Natural function space for Maxwell equations: Find A ∈ H(curl) such that∫Ω

µ−1 curlA curl v +

∫Ω

(iσω − εω2)Av =

∫jv ∀ v ∈ H(curl)


Contents

• Introduction

• Hybrid Discontinuous Galerkin Method

• Finite Elements for H(div) and H(curl)

• Tangential-continuous finite elements for elasticity

• Normal-continuous finite elements for Stokes


Hybrid Discontinuous Galerkin (HDG) Method

Model problem: −∆u = f

A mesh consisting of elements and facets (= edes in 2D and faces in 3D)

T = T F = F

Goal: Approximate u with piece-wise polynomials on elements and additional polynomials on facets:

uN ∈ P p(∪T ) λN ∈ P p(∪F )

Joachim Schoberl Hybrid Discontinuous Galerkin (HDG)

HDG - Derivation

Exact solution u, traces on element boundaries: λ := u|∪F

Integrate against discontinuous test-functions v ∈ H1(∪T ), element-wise integration by parts:

∑T

∫T

∇u∇v −∫∂T

∂u

∂nv

=

∫Ω

fv

Use continuity of ∂u∂n, and test with single-valued functions µ ∈ L2(∪F ):

∑T

∫T

∇u∇v −∫∂T

∂u

∂n(v − µ)

=

∫Ω

fv

Use consistency u = λ on ∂T to symmetrice, and stabilize ...

∑T

∫T

∇u∇v −∫∂T

∂u

∂n(v − µ)−

∫∂T

∂v

∂n(u− λ) + α (u− λ, v − µ)j,∂T

=

∫Ω

fv

Dirichlet b.c.: Imposed on λ, Neumann b.c.:∫

ΓNgµ


Interior penalty method

Stabilization with α suff large

α (u− λ, v − µ)j,∂T =αp2

h(u− λ, v − µ)L2(∂T )

Norm:‖(u, λ)‖21,HDG := ‖∇u‖2L2(T ) + ‖u− λ‖2j,T

Stability is proven by Young’s inequality and inverse inequality ‖∂u∂n‖2L2(∂T ) ≤ cinv

p2

h ‖∇u‖2L2(T ):

AT (u, λ;u, λ) = ‖∇u‖2L2(T ) − 2

∫∂T

∂u

∂n(u− λ)︸︷︷︸

≤√

cinvα ‖∇u‖

2L2(T )

+√cinvα

p2

h ‖u−λ‖2L2(∂T )

+αp2

h‖u− λ‖2L2(∂T )

' ‖(u, λ)‖21,HDG

for α > cinv.


Bassi-Rebay type method

Stabilization term isα (u− λ, v − µ)j,∂T = α

(r(u− λ), r(v − µ)

)L2(T )

with lifting operator r : P p(FT )→ [P p(T )]d such that

(r(u− λ), σ)L2(T ) = (u− λ, σn)L2(∂T ) ∀σ ∈ [P p(T )]d

The corresponding jump-norm is

‖u− λ‖j,∂T = supσ∈[P p(T )]d

(u− λ, σn)L2(∂T )

‖σ‖L2(T )

Stability for any α > 1:

AT (u, λ;u, λ) = ‖∇u‖2L2(T ) − 2

∫∂T

∂u

∂n(u− λ)︸︷︷︸

≤‖∇u‖L2(T ) supσ∈[Pp]d

∫∂T σn(u−λ)

‖σ‖L2(T )

+α‖u− λ‖2j,T

' ‖(u, λ)‖21,HDG


Error estimates

Follows from consistency and discrete stability:

‖(u− uN , u− λN)‖1,HDG infvN ,µN

‖∇(u− vN)‖L2(T ) + ‖uN − λN‖j + ‖∂nu− ∂nuN‖j∗

pγhs

ps‖u‖H1+s

• for 1 ≤ s ≤ p

• with γ = 1/2 or γ = 0 depending on mesh-conformity, and jump-term.


Convection - Diffusion Problems

−ε∆u+ b · ∇u = f in ∂Ω

u = 0 on ∂Ω

HDG Formulation:

Ad(u, λ; v, µ) +Ac(u, λ; v, µ) =

∫fv

with diffusive term Ad(., .) from above and upwind-discretization for convective term

Ac(u, λ; v, µ) =∑T

−∫bu · ∇v +

∫∂T

bnu/λv

with upwind choice

u/λ =

λ if bn < 0, i.e. inflow edgeu if bn > 0, i.e. outflow edge

assuming div b = 0. Then Ac(u, λ;u, λ) ≥ 0 (and inf − sup stability)


Results for 1D

−εu′′ + u′ = 1, u(0) = u(1) = 0

HDG Discretization:left: ε = 10−2

right: ε = 10−4

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

eps = 0.01

n = 10n = 100

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

eps = 0.0001

n = 10n = 100

conforming elements withSUPG stabilization

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

eps = 0.01

n = 10n = 100

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

eps = 0.0001

n = 10n = 100


Relation to standard Interior Penalty DG method

DG - spaceVN := P p(∪T )

Bilinearform

ADG(u, v) =∑T

∫T

∇u∇v −∫∂T

∂u

∂n[v]−

∫∂T

∂v

∂n[u] +

αp2

h

∫∂T

[u][v]

Hybrid DG has

• even more unknowns, but less matrix entries

• allows element-wise assembling

• allows static condensation of element unknowns

Hybridization of standard DG methods [Cockburn+Gopalakrishnan+Lazarov]


Relation to classical hybridization of mixed methods

First order systemAσ −∇u = 0 and div σ = −f

Mixed method: Find σ ∈ H(div) and u ∈ L2 such that∫Aστ −

∫div τ u = 0 ∀ τ ∈ H(div)∫

div σ v = −∫fv ∀ v ∈ L2

Breaking normal-continuity of σn, and reinforcing it by another Lagrange parameter [Arnold-Brezzi, 86]

Find σ ∈ H(div), u ∈ L2, and λ ∈ L2(∪F ) such that∫Aστ +

∑T

∫T

div τ u+∑F

∫F

[τn]λ = 0 ∀ τ ∈ H(div)∑T

∫T

div σ v = −∫fv ∀ v ∈ L2∑

F

∫F

[σn]µ = 0 ∀µ ∈ L2(∪F )

Allows to eliminate σ (and also u) leading to a coercive system in u and λ (or, only λ).


Comparison to mixed hybrid system

HDG method needs facet variable of one order higher ???

λ ∈ P p−1(∪F ) is enough when inserting a projector:

AHDG(u, λ; v, µ) =∑T

∫T

∇u∇v −∫∂T

∂u

∂n(v − µ)

−∫∂T

∂v

∂n(u− λ) +

αp2

h

∫∂T

Πp−1(u− λ) Πp−1(v − µ)

Implementation of the projector by an EAS - like method.


How to solve ?

Standard DG

κC−1ASMA ' p

2

for element-by-element Schwarzpreconditioner CASM plus coarse grid[Antonietti+Houston,11]

Hybrid DGwith facet Schur-complement S

κC−1ASMS ' (log p)γ

for facet-by-facet Schwarz preconditionerCASM plus coarse grid

Joachim Schoberl Preconditioning

Trace norms inequality

For λ ∈ P p(F ) define semi-norm and norm

|λ|2F := infu∈P p

‖∇u‖2L2(T ) + ‖u− λ‖2j,F

‖λ‖2F,0 := inf

u∈P p

‖∇u‖2L2(T ) + ‖u− λ‖2j,F + ‖u− 0‖2j,∂T\F

mimic | · |H1/2(F ) and ‖ · ‖

H1/200 (F )

.

Theorem: For λ ∈ P p(F ) with∫Fλ = 0 there holds

‖λ‖2F,0 (log p)γ|λ|2F with γ = 3

• if T is a trig, quad, or hex, and ‖ · ‖j is IP or BR

• if T is a tet, and ‖ · ‖j is BR

From the trace norms inequality we get immediately condition number estimates for Schwarz methods andBDDC preconditioners


Condition numbers for BDDC preconditioner

Laplace equation, mesh consisting of 184 tetrahedra, HDG discretization

50

100

150

200

250

300

1 10

cond

ition

num

ber

polynomial order

BR-elementIP al=10IP al=20IP al=40

• Bassi-Rebay with α = 1.5 (proven to be O(log3 p))

• interior penalty with α = 10, 20, 40 (only O(p) is proven)


Mixed Continuous / Hybrid Discontinuous Galerkin method

Vector-valued spaces with partial continuity and partial components on facets:

VT ,n = v ∈ [P p(∪T )]d : [vn] = 0 VT ,τ = v ∈ [P p(∪T )]d : [vτ ] = 0

VF,n = v ∈ [P p(∪F )]d : vτ = 0 VF,τ = v ∈ [P p(∪F )]d : vn = 0

H(curl) - based formulation for elasticity: Find u ∈ VT ,τ and λ ∈ VF,n such that

Aτ(u, λ; v, µ) =

∫fv ∀ v ∈ VT ,τ ∀µ ∈ VF,ν

Aτ(u, λ; v, µ) =∑T

∫T

Dε(u) : ε(v)−∫∂T

(Dε(u))nn(v − µ)n

−∫∂T

(Dε(v))nn(u− λ)n +αp2

h

∫∂T

(u− λ)n(v − µ)n

Or, vice versa ...


The de Rham Complex

H1 ∇−→ H(curl)curl−→ H(div)

div−→ L2⋃ ⋃ ⋃ ⋃Wh

∇−→ Vhcurl−→ Qh

div−→ Sh

satisfies the exact sequence property

range(∇) = ker(curl)

range(curl) = ker(div)

on the continuous and the discrete level.

Important for stability, error estimates, preconditioning, ...

Joachim Schoberl Finite elements for H(curl) and H(div)

Construction of high order H(curl) and H(div) finite elements

• [Dubiner, Karniadakis+Sherwin] H1-conforming shape functions in tensor product structure→ allows fast summation techniques

• [Webb] H(curl) hierarchical shape functions with local exact sequence propertyconvenient to implement up to order 4

• [Demkowicz+Monk] Based on global exact sequence property construction of Nedelec elements ofvariable order (with constraints on order distribution) for hexahedra

• [Ainsworth+Coyle] Systematic construction of H(curl)-conforming and H(div)-conforming elements ofarbitrarily high order for tetrahedra

• [JS+Zaglmayr] Based on local exact sequence property and by using tensor-product structure weachieve a systematic strategy for the construction of H(curl)-conforming hierarchical shape functionsof arbitrary and variable order for common element geometries (segments, quadrilaterals,triangles, hexahedra, tetrahedra, prisms, pyramids).[COMPEL, 2005], PhD-Thesis Zaglmayr 2006


Hierarchical V EFC basis for H1-conforming Finite Elements

The high order elements have basis functions connected with the vertices, edges, (faces, ) and cell of themesh:

Vertex basis function Edge basis function p=3 Inner basis function p=3

This allows an individual polynomial order for each edge, face, and cell..


High-order H1-conforming shape functions in tensor product structure

Exploit the tensor product structure of quadrilateral elements to build edge and face shapes

Family of orthogonal polynomials (P 0k [−1, 1] )2≤k≤p vanishing in ±1.

ϕFi j(x, y) = P 0i (x)P 0

j (y),

ϕE1i (x, y) = P 0

i (x) 1−y2 .

Tensor-product structure for triangle [Dubiner, Karniadakis+Sherwin]:

Collapse the quadrilateral to the triangle by x→ (1− y)x

ϕE1i (x, y) = P 0

i ( x1−y) (1− y)i

ϕFi j(x, y) = P 0i (

x

1− y)(1− y)i︸︷︷︸

ui(x,y)

Pj(2y − 1)y︸︷︷︸vj(y)

Remark: Implementation is free of divisions


The deRham Complex tells us that ∇H1 ⊂ H(curl), as well for discrete spaces ∇W p+1 ⊂ V p.

Vertex basis function Edge basis function p=3 Inner basis function p=3


The deRham Complex tells us that ∇H1 ⊂ H(curl), as well for discrete spaces ∇W p+1 ⊂ V p.

Vertex basis function

y∇

Edge basis function p=3

y∇

Inner basis function p=3

y∇

∇WVi ⊂ VN0 ∇W p+1Ek

= V pEk ∇W p+1Fk⊂ V pFk

Joachim Schoberl Finite elements for H(curl) and H(div) Page 26

H(curl)-conforming face shape functions with ∇W p+1F ⊂ V pF

We use inner H1-shape functions spanning W p+1F ⊂ H1 of the structure

ϕF,∇i,j = ui(x, y) vj(y).

We suggest the following H(curl) face shape functions consisting of 3 types:

• Type 1: Gradient-fields

ϕ F,curl1, i,j = ∇ϕF,∇i,j = ∇(ui vj) = ui∇vj + vj∇ui

• Type 2: other combination

ϕ F,curl2, i,j = ui∇vj− vj∇ui

• Type 3: to achieve a base spanning VF (p− 1) lin. independent functions are missing

ϕ F,curl3, j = N0(x, y) vj(y).

Similar in 3D and for H(div).


Localized exact sequence property

We have constructed Vertex-Edge-Face-Cell shape functions satisfying

WVh, p+1=1

∇−→ V N0h

curl−→ QRT 0h

div−→ Sh, 0

WEpE+1

∇−→ V EpE

WFpF+1

∇−→ V FpFcurl−→ QFpF−1

WCpC+1

∇−→ V CpCcurl−→ QCpC−1

div−→ SCpC−2.

Advantages are

• allows arbitrary and variable polynomial order on each edge, face and cell

• Additive Schwarz Preconditioning with cheap N0 − E − F − C blocks gets efficient

• Reduced-basis gauging by skipping higher-order gradient bases functions

• discrete differential operators B∇, Bcurl, Bdiv are trivial


Magnetostatic BVP - The shielding problem

Simulation of the magnetic field induced by a coil with prescribed currents:

Electromagnetic shielding problem: magnetic field, p=5

Absolute value of magnetic flux, p=5

... prism layer in shield, unstructured mesh (tets, pyramids) in air/coil.

p dofs grads κ(C−1A) iter solvertime

4 96870 yes 34.31 37 24.9 s4 57602 no 31.14 36 6.6 s

7 425976 yes 140.74 63 241.7 s7 265221 no 72.63 51 87.6 s


Application: Simulation of eddy-currents in bus bars

Full basis for p = 3 in conductor, reduced basis for p = 3 in air


Elasticity: A beam in a beam

Reenforcement with E = 50 in medium with E = 1.

HDG FEM, p = 3 Primal FEM, p = 3

Joachim Schoberl Mixed elasticity Page 31

Tangential displacement - normal normal stress constinuous mixed method

[Phd thesis Astrid Sinwel 09 (now Astrid Pechstein)], [A. Pechstein + JS 2011]

Mixed elements for approximating displacements and stresses.

• tangential components of displacement vector

• normal-normal component of stress tensor

Triangular Finite Element:

u

σ

τ

nn

Tetrahedral Finite Element:

u

σnn

τ


The quadrilateral element

Dofs for general quadrilateral element:

uτ

σnn

Thin beam dofs (σnn = 0 on bottom and top):

Beam stretching components:

σnnu τ mean value mean value

Beam bending components:

uτverticaldeflection

momentbendingrotation


Hellinger Reissner mixed methods for elasticity

Primal mixed method:

Find σ ∈ Lsym2 and u ∈ [H1]2 such that∫Aσ : τ −

∫τ : ε(u) = 0 ∀ τ

−∫σ : ε(v) = −

∫f · v ∀ v

Dual mixed method:

Find σ ∈ H(div)sym and u ∈ [L2]2 such that∫Aσ : τ +

∫div τ · u = 0 ∀ τ∫

div σ · v = −∫f · v ∀ v

[Arnold+Falk+Winther]


Reduced Symmetry mixed methods

Decomposeε(u) = ∇u+ 1

2 Curlu = ∇u+ ω

with Curlu = 2 skew(∇u) =(∂xiuj − ∂xjui

)i,j=1,...d

Impose symmetry of the stress tensor by an additional Lagrange parameter:

Find σ ∈ [H(div)]d, u ∈ [L2]d, and ω ∈ Ld×d,skew2 such that∫Aσ : τ +

∫udiv τ +

∫τ : ω = 0 ∀ τ∫

v div σ = −∫fv ∀ v∫

σ : γ = 0 ∀ γ

The solution satisfies u ∈ L2 and ω = Curlu ∈ Ld×d,skew2 , i.e.,

u ∈ H(curl)

Arnold+Brezzi, Stenberg,... 80s


Choices of spaces

∫div σ · u understood as

〈div σ, u〉H−1×H1 = −(ε(u), σ)L2 (div σ, u)L2

Displacement

u ∈ [H1]2 u ∈ [L2]2

continuous f.e. non-continuous f.e.

Stress

σ ∈ Lsym2 σ ∈ H(div)sym

non-continuous f.e. normal continuous (σn) f.e.

The mixed system is well posed for all of these pairs.


Choices of spaces

∫div σ · u understood as

〈div σ, u〉H−1×H1 = −(ε(u), σ)L2 〈div σ, u〉H(curl)∗×H(curl) (div σ, u)L2

Displacement

u ∈ [H1]2 u ∈ H(curl) u ∈ [L2]2

continuous f.e. tangential-continuous f.e. non-continuous f.e.

Stress

σ ∈ Lsym2 σ ∈ Lsym2 ,div div σ ∈ H−1 σ ∈ H(div)sym

non-continuous f.e. normal-normal continuous (σnn) f.e. normal continuous (σn) f.e.

The mixed system is well posed for all of these pairs.


The TD-NNS-continuous mixed method

Assuming piece-wise smooth solutions, the elasticity problem is equivalent to the following mixed problem:Find σ ∈ H(div div) and u ∈ H(curl) such that∫

Aσ : τ +∑T

∫T

div τ · u−∫∂Tτnτuτ

= 0 ∀ τ∑

T

∫T

div σ · v −∫∂Tσnτvτ

= −

∫f · v ∀ v

Proof: The second line is equilibrium, plus tangential continuity of the normal stress vector:∑T

∫T

(div σ + f)v +∑E

∫E

[σnτ ]vτ = 0 ∀ v

Since the space requires continuity of σnn, the normal stress vector is continuous.Element-wise integration by parts in the first line gives∑

T

∫T

(Aσ − ε(u)) : τ +∑E

∫E

τnn[un] = 0 ∀ τ

This is the constitutive relation, plus normal-continuity of the displacement. Tangential continuity of thedisplacement is implied by the space H(curl).


Reissner Mindlin Plates

Energy functional for vertical displacement w and rotations β:

‖ε(β)‖2A−1 + t−2‖∇w − β‖2

MITC elements with Nedelec reduction operator:

‖ε(β)‖2A−1 + t−2‖∇w −Rhβ‖2

Mixed method with σ = A−1ε(β) ∈ H(div div), β ∈ H(curl), and w ∈ H1:

L(σ;β,w) =1

2‖σ‖2A + 〈div σ, β〉 − t−2‖∇w − β‖2

Joachim Schoberl Thin structures Page 38

Reissner Mindlin Plates and Thin 3D Elements

Mixed method with σ = A−1ε(β) ∈ H(div div), β ∈ H(curl), and w ∈ H1:

L(σ;β,w) = ‖σ‖2A + 〈div σ, β〉 − t−2‖∇w − β‖2

Reissner Mindlin element:

σ

τ

nn

β

w

3D prism element:

σnn uτ


Anisotropic Estimates

Thm: There holds∑T

‖ε(u− uh)‖2T +∑F

h−1op ‖[un]‖2F + ‖σ − σh‖2 ≤ c

hmxy‖∇mxyε(u)‖+ hmz ‖∇mz ε(u)‖

2

Proof: Stability constants are robust in aspect ratio (for tensor product elements)

Anisotropic interpolation estimates (H1: Apel). Interpolation operators commute with the strain operator:

‖ε(u−Qu)‖L2 = ‖(I − Q)ε(u)‖L2

hmxy‖∇mx εxy,z(u)‖0 + hmz ‖∇mz εxy,z(u)‖L2

[A. Pechstein + JS, 2011]


For Hot Days ...

Geometry

Deformed geometry, stress σxx

Displacement uy

Interior stress


Contact problem with friction

Undeformed bearStress, component σ33


Shell structure

R = 0.5, t = 0.005σ ∈ P 2, u ∈ P 3

stress component σyy


Hybridization: Implementation aspects

Both methods are (essentially) equivalent:

• Classical hybridization of mixed method:

Introduce Lagrange parameter λn to enforce continuity of σnn. Its meaning is the displacement innormal direction.

• Continuous / hybrid discontinuous Galerkin method:

Displacement u is strictly tangential continuous, HDG facet variable (= normal displacement) enforcesweak continuity of normal component.

Anisotropic error estimates from mixed methods can be applied for HDG method !


Continuous / hybrid discontinuous Galerkin method for Stokes

(Thesis C. Lehrenfeld 2010, RWTH)

H(div) - based formulation for Stokes:

Find u ∈ VT ,n ⊂ H(div), λ ∈ VF,τ and p ∈ P p−1(T ) such that

An(u, λ; v, µ) +∫

Ωdiv v q =

∫fv ∀ (v, µ)∫

div u q = 0 ∀ q

Provides exactly divergence-free discrete velocity field u

LBB is proven by commuting interpolation operators for de Rham diagram

[Cockburn, Kanschat, Schotzau 2005]

Joachim Schoberl Navier Stokes Equations Page 45

H(div)-conforming elements for Navier Stokes

∂u

∂t− div(2νε(u)− u⊗ u− pI) = f

div u = 0

+b.c.

Fully discrete scheme, semi-implicit time stepping:

(1

τM +Aν)u+BT p = f +

1

τMu−Ac(u)

Bu = 0

• u is exactly div-free ⇒ non-negative convective term∫u∇vv ≥ 0

• stability for kinetic energy ( ddt‖u‖20 1

ν‖f‖2L2

)

• convective term by upwinding

• allows kernel-preserving smoothing and grid-transfer for fast iterative solver


The de Rham Complex

H1 ∇−→ H(curl)curl−→ H(div)

div−→ L2⋃ ⋃ ⋃ ⋃Wh

∇−→ Vhcurl−→ Qh

div−→ Sh

For constructing high order finite elements

Whp = WL1 + spanϕWh.o.Vhp = VN0 + span∇ϕWh.o.+ spanϕVh.o.

Qhp = QRT 0 + spancurlϕVh.o.+ spanϕQh.o.Shp = SP0 + spandivϕSh.o.

Allows to construct high-order-divergence free elements v ∈ BDMk : div v ∈ P0


Flow around a disk, 2D

Re = 100, 5th-order elements

Boundary layer mesh around cylinder:


Flow around a disk, 2D

Re = 1000:

Re = 5000:


Flow around a cylinder, Re = 100


Concluding Remarks

• Hybrid DG is a simple and efficient hp - discretization scheme

• Robust anisotropic elements for linear elasticity

• Exactly divergence free finite elements for incompressible flows

Ongoing work:

• Operator splitting time integration

• Preconditioning (BDDC element-level domain decomposition)

• MPI-based Parallelization, GPU implementation of explicit time-stepping methods

Open source software on sourceforge:

• Netgen/NGSolve : Mesh generator and general purpose finite element code

• NGS-flow : CFD module for Netgen/NGSolve


Hybrid Discontinuous Galerkin Methods for Fluid Dynamics ...

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